A. W. F. Edwards gave a construction to higher numbers of sets that features some symmetries. His construction is achieved by projecting the Venn diagram onto a sphere. Three sets can be easily represented by taking three hemispheres at right angles (*x*≥0, *y*≥0 and *z*≥0). A fourth set can be represented by taking a curve similar to the seam on a tennis ball which winds up and down around the equator. The resulting sets can then be projected back to the plane to produce “cogwheel” diagrams with increasing numbers of teeth. These diagrams were devised while designing a stained-glass window in memoriam to Venn.

### Other Diagrams

Edwards's Venn diagrams are topologically equivalent to diagrams devised by Branko Grünbaum which were based around intersecting polygons with increasing numbers of sides. They are also two-dimensional representations of hypercubes.

Smith devised similar *n*-set diagrams using sine curves with equations *y*=sin(2^{i}*x*)/2^{i}, 0≤i≤*n*-2.

Charles Lutwidge Dodgson (a.k.a. Lewis Carroll) devised a five set diagram.

## Classroom Use

Venn diagrams are often used by teachers in the classroom as a graphic organizer, a mechanism to help students compare and contrast two or three “sets” of ideas. Characteristics of each set of ideas are listed in each section of the diagram, with shared characteristics listed in the overlapping sections. Simple Venn diagrams are introduced to students as early as kindergarten, and are used to help students organize their thoughts before writing about them.

In Indian schools the basic Venn diagrams are taught using Indian rupee coins.

## Example

Sets A and BThe orange circle (set *A*) might represent, for example, all living creatures that are two-legged . The blue circle, (set *B*) might represent living creatures that can fly. The area where the blue and orange circles overlap (which is called the *intersection*) contains all living creatures that can fly **and** which have two legs-for example, parrots. (Imagine each separate type of creature as a point somewhere in the diagram.)

Humans and penguins would be in the orange circle, in the part that does not overlap with the blue circle. Mosquitoes have six legs, and fly, so the point for mosquitoes would be in the part of the blue circle that does not overlap with the orange one. Things that do not have two legs and cannot fly (for example, whales and rattlesnakes) would all be represented by points outside both circles. Technically, the Venn diagram above can be interpreted as "the relationships of set *A* and set *B* that may have some (but not all) elements in common."

The combined area of sets *A* and *B* is called the “union” of sets *A* and *B.* The union in this case contains all things that either have two legs, can fly, or both.

The area in both *A* and *B,* where the two sets overlap, is defined as *A*∩*B,* that is, *A* **intersected** with *B.* The intersection of the two sets is not empty, because the circles overlap, i.e. there are creatures that are in **both** the orange and blue circles.

Sometimes a rectangle, called the Universal set, is drawn around the Venn diagram to represent the space of all possible things under consideration. As mentioned above, a whale would be represented by a point that is not in the union, but is in the Universe (of living creatures, or of all things, depending on how one chose to define the Universe for a particular diagram).

## See also

- Boolean algebra

## Notes

- ↑ Ruskey, F., Venn Diagrams. Retrieved October 25, 2007.
- ↑ Weisstein, Eric W., Venn Diagram. Retrieved October 25, 2007.
- ↑ Henderson, D.W., "Venn diagrams for more than four classes,"
*American Mathematical Monthly,*70, (1963), p. 424-426. - ↑ Ruskey, Frank, Savage, Carla D., and Wagon, Stan, The Search for Simple Symmetric Venn Diagrams. Retrieved October 25, 2007.

## References

- Cieutat, Victor J., Krimerman, Leonard I., and Elder S. Thomas.
*Traditional logic and the Venn diagram; a programed introduction.*San Francisco: Chandler Pub. Co. 1969. - Edwards, A. W. F.
*Cogwheels of the mind : the story of Venn diagrams.*Baltimore: Johns Hopkins University Press. 2004. ISBN 0801874343 - McCarthy, J.F., and Krishnamoorthy, M.S. "Venn Diagram Construction of Internet Chatroom Conversations."
*Computer Science.*(3975):535-541. 2006. - Stewart, Ian.
*Another fine math you've got me into.*New York: W.H. Freeman. 1992. ISBN 0716723425 - Venn, John. "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings."
*Dublin Philosophical Magazine and Journal of Science.*volume 9, 59, 1880. p. 1-18.

## External Links

All links retrieved January 19, 2016.

- Bogomolny, Alexander. Venn Diagrams.
- Bogomolny, Alexander. Venn Diagrams (Click).
- Dunham, William. Lewis Carroll's Logic Game.
- Johnston, Russell. Illustrating Formal Logic with Exclusion Diagrams.
- Rodgers, Peter. Applet for Drawing 3 Set Area-Proportional Venn Diagrams.
- Ruskey, F., and Weston, M. A Survey of Venn Diagrams.
- SourceForge. Venn Diagrams.